How to Choose the Right Control Chart for Statistical Process Control

    For characteristics which are evaluated with variable data, AIAG’s SPC manual recommended 4 types of control charts:

1. Average and Range Chart (X ̅-R Chart)
2. Average and Standard Deviation Chart (X ̅-s Chart)
3. Median and Rang Chart (X ̃-R Chart)
4. Individuals and Moving Range Chart (X-MR Chart)

    Arising from the 4 options, a question that many SPC beginners may have is: which control chart should I use for my process? The answer can be found in P.177 of the 4th edition of the SPC manual, but it is very brief. Below is a more detailed explanation about the selection procedure of the right control chart:

    First, let’s take a look at the X ̅-R Chart, which is probably the most popular chart. For this chart, the control limits of X ̅ and R are calculated with the following equations:

where R ̅ is the average of ranges of individual subgroup. From the above equations, it can be seen that when R ̅ is 0,and. The control limits make no sense in such case. That means if one needs to use X ̅-R chart, there must exist detectable within-subgroup variation. The above equations also show that the width between UCL and LCL of X ̅  is. In order not to see too many undesired out of control signals, the X ̅-R Chart should only be used when the between-subgroup variation is insignificant compared with within-subgroup variation. Otherwise, the future data point of X ̅ may easily go out the control limits due to big between-subgroup variation (Note: generally, big between-subgroup variation, compared with within-subgroup variation, is not accepted as it indicates the existence of special causes. But if both Cpk and Ppk are significantly larger than the requirement, e.g. Cpk=8, Ppk=5, it may be accepted). As a summary, to use X ̅-R chart, the within-subgroup variation must not be 0 and the between-subgroup variation must be insignificant compared with within-subgroup variation.

    Here is an example when the X ̅-R chart should not be used: The process is to produce a chemical solution by adding solvent into the distilled water. It is produced one bath a time, and one bath is one lot. The monitored characteristic is the composition of the solution. As the solution is homogenous, taking several samples from one bath as one subgroup gives no within-subgroup variation. On the other hand, the difference between different lots may be more significant, as the amount of solvent added to each bath may differ. So using X ̅-R chart is inappropriate for this process.

  In the above case, one should use the X-MR chart. The X-MR chart requires no subgroups (or you may consider that the subgroup size is 1). Individual sample is taken, and the difference between consecutive samples is calculated to estimate the process variation and the control limits, as in the following equations:

where (MR) ̅ is the average of the differences between all consecutive samples.

    X ̅-s chart is close to X ̅-R chart. The key difference is that the process variation in this chart is estimated with the average standard deviations, instead of average ranges, of the subgroups, and the control limits are calculated with the following equations:

where s ̅ is the average of the standard deviations of individual subgroup. Similar to X ̅-R Chart, X ̅-s chart should only be used when within-subgroup variation is detectable, and the between-subgroup variation is insignificant compared with within-subgroup variation.

Since X ̅-s chart is very close to X ̅-R chart, when should they be used respectively? In fact, X ̅-s is always preferred over X ̅-R chart, as the process variation σ ̂_X ̅  is better estimated with average standard deviation of subgroups than average range of subgroups. One may consider using X ̅-R chart only if the SPC cannot be done with computer and hence calculation of standard deviation is inconvenient. But if the subgroup size is 9 or more, the estimation of σ ̂_X ̅  with average range of subgroups becomes too poor, and X ̅-s chart should be used in this case, even the calculation of standard deviation is difficult.

    X ̃-R chart is also close to X ̅-R chart. But in this chart, median, instead of average, of each subgroup is monitored. Comparing X ̃-R chart with X ̅-R chart, the latter is preferred, since for process control, one cares about the location of the process average, which is better estimated with X ̅ than X ̃. X ̃-R chart should be used only if the calculation of subgroup average is not convenient, e.g. when doing SPC on a paper.

    As a summary, when choosing the right control charts, one needs to follow the below priority:

1. Average and Standard Deviation Chart (X ̅-s Chart)
2. Average and Range Chart (X ̅-R Chart)
3. Median and Rang Chart (X ̃-R Chart)
4. Individuals and Moving Range Chart (X-MR Chart)

i.e. X ̅-s chart should be always considered first, but if the subgroup size is less than 9 and the calculation of standard deviation is not convenient, X ̅-R chart can be used as an alternative. And if the calculation of subgroup average is also not convenient, then X ̃-R chart can be used instead of X ̅-R chart. All the above control charts require that subgroups with detectable within-subgroup variation can be obtained, and the between-subgroup variation is insignificant compared with within-subgroup variation. If it’s not possible to obtain such subgroups, one can then use X-MR chart.

How to Use SPC for Process Control and Improvement

      SPC is a powerful tool for process control and improvement, but if it’s not properly used, it may cause extra burden to the owners of processes, but bring them with little benefit. To do effective process control and improvement with SPC, one needs to follow the three steps as described below:

1. Process Release

      This is done during R&D stage for transferring the process to mass production. A process must be evaluated and can only be released for mass production when it satisfies two conditions:

• The process is under statistical control. In other words, there is no special cause existed and the process is stable and predictable. The distribution of products produced by this process will be the same in the future. 
• The indicators of the process capability and performance meet the customer or internal requirements (e.g. many customers require Cpk>1.67).

      Now let’s see how the evaluation on the process should be done:

      1) First, collect >= 25 subgroups with a total of >=100 samples of the controlled parameter. The variation within a subgroup should be minimized, so the samples of the same subgroup are suggested to be taken from the same batch or lot, with the least variation in the impacting factors of the process (e.g. materials used, operators, equipment, processing parameters, etc). On the other hand, the variation between subgroups should be maximized. So a subgroup can be taken when the shift is changed, the material lot is changed, or the production line is restarted, etc.

      2) Calculate the trial control limits with the data points collected. Plot them in the control charts. Identify any subgroup which signals the existence of a special cause. Exclude these subgroups and repeat this step (more subgroups may be collected to replace the excluded ones if the total number of subgroups falls within 25). Actions should be taken to remove the special cause if it continuously exists in the process.

      3) When all the subgroups with special causes are excluded from the control charts, use the reaming subgroups to calculate the control limits and the indicators of the process capability and performance, including Cp, Cpk, Pp and Ppk, and see whether these indicators meet the requirements (e.g. whether Cpk>1.67).

      4) If not, take actions on the common cause to improve the process until the indicators meet the requirements.

      5) If yes, check the difference between Cp and Cpk. If there exists a big difference, it means the process average is significantly shifted from the process target, and a significant part of the process capability is wasted due to the shift . Actions should be taken to bring the process average close to the target.

      6) Also, it’s suggested to check the difference between Cpk and Ppk. If the difference is significant, it indicates the between-subgroup variation is significant, special causes exist and the process is not actually in statistical control. Actions should be taken to remove the special causes.

      After all the above steps, special causes originally existed in the process are removed (i.e. the process is now statistically controlled) and all the process capability and performance meet the requirements. The process can now be released for mass production.

      After the process is released, the control limits established after the above steps should be continually employed during the mass production. They should not be changed unless there’s intentional change in the process or the reason for the change is clear.

2. Process Monitoring

      With the control limits as established during process releasing, the process owner continues the data collection of subgroups, with the defined sampling frequency, sampling scheme and size of subgroups, and monitors:

• Whether the new subgroups indicate the occurrence of any special cause (the rules to identify the special cause can be found in P.75 of AIAG's SPC manual);
• Whether the indicators of process capability and performance fail to meet the requirements; 
• Whether the difference between Cp and Cpk or Cpk and Ppk is significant. 

      If the answer is YES to any of the above question, actions should be taken as discussed in Step 3)-6) in Step 1 Process Release.

3. Process Improvement

      Even the answers for the three questions in Process Monitoring are all NO, one should never stop there. One needs to study the process and take actions to improve the process, to increase the process capability and performance. The improvement of process requires actions on the common causes, and usually requires the resource support from the management team.

      Once the improvement actions have been taken, one needs to go back to Step 1 Process Release, do the evaluation on the process and start over again. As a most significant result from the process improvement, new and tighter control limits are usually established.

The Difference Between Cpk and Ppk

      What is the difference between Cpk and Ppk? How should they be properly used for process control and improvement? These are probably questions confusing many SPC users. This article will try to answer these questions.

      First, let’s compare the difference between Cpk and Ppk. Cpk is an indicator for the capability of the process, the best that the process can achieve, while Ppk is an indicator for the actual performance of the process. A process with, say, Cpk=1.67 may not show its full capability and only shows a Ppk below 1.67, say, 1.50. An analogy to this situation is a student who’s really smart and has the ability to become the top 1 student in the class (his capability), but it’s not necessary that he’ll become the top 1 student (his performance), for reasons such as less devotion than other students.

      Second, let’s see why Cpk is an indicator of the process capability, while Ppk is an indicator of the process performance.

      To answer this question, we must keep it mind that the first step for SPC control is to adjust the process and ensure it is statistically controlled. Meanwhile, for the sake of easy formulation, let’s assume that the process average coincides with the process target. In such a case

      In the above Equation I, (USL-LSL) is the accepted tolerance of the process, and σc the standard deviation of the measured samples (please be noted that it is the standard deviation of all the measured samples, not the average of the subgroups), an indicator of the total process variation. The higher their ratio (i.e. Cpk) is, the lower the chance is to produce a nonconforming product. For a process with a normal distribution and with the process target coinciding with the process average, the correspondence between Cpk and the defective rate (in terms of PPM) is given the following table:

      From the table, it can be seen that a higher Cpk results in a lower defective rate, indicating a better ability of the process in producing conforming products.

      In Equation I, σc is calculated with the following equation:

where Rbar is the average of the ranges of individual subgroups. Combining Equation I and Equation II, one can see that the capability of a process Cpk is determined by variation within each subgroup.

      At this point, one may wonder why the total process variation σc is only determined by the within-subgroup variation. How about the between-subgroup variation? To understand that, we must mention the precondition to evaluate the process capability again: the process is statistically controlled. For such a process, the between-subgroup variation is negligible (according to P.131 in the second revision of AIAG SPC’s manual, the between-subgroup variation is 0 for a statistically controlled process). Therefore, the total process variation, and hence Cpk, is only determined by the within-subgroup variation (please refer to the below image).

      In real life, however, a process cannot always be adjusted to statistically controlled state even there is no special cause detected in the control chart. Also, even if the process is in statistical control at the moment of evaluation, a special cause can occur at any future moment and take the process away from statistically controlled state. Therefore, for an actual process, its performance hardly shows its capability as indicated by Cpk. The total variation of the actual process consists of not only the within-subgroup variation, but also the between subgroup variation. The performance of the process, Ppk, is calculated with the following equation

where σp is total variation of the actual process, and it is the combined result of within and between-subgroup variation (please refer to the image below), and is calculated with the following equation:

where Xi is the data point and n is the total number of the sampled data.

      As a summary, Cpk is calculated for a process which is under statistical control. The between-subgroup variation is 0, and the process variation comes from within-subgroup variation only. The process in such state is the best performance that a process can achieve, so Cpk represents the process capability. On the other hand, Ppk is calculated for actual process which is not necessarily under statistical control. The process variation consists of both within and between subgroup variations. Therefore, Ppk reflects the actual performance of the process. Ppk can never be larger than Cpk.

      Finally, after understanding the difference of Cpk and Ppk, let’s consider how these two indicators should be used for process control and improvement. There’s no question that these two indicators should be monitored to meet customer or internal requriements (e.g. Cpk >1.67). Meanwhile, these two indicators should be monitored together and their difference is evaluated. A big difference between Cpk and Ppk indicates large between-subgroup variation, meaning special cause may exist in the process and actions should be taken to remove the special cause. Cpk and Ppk should never walk alone.